Advanced Synthetic Control Methods

Lee Kennedy-Shaffer, PhD

2024-06-18

Key Ideas

  • Allow for staggered adoption

  • Reduce interpolation

  • Account for exposed series outside of the convex hull

  • Incorporate control series on different outcomes/scales from treated series

Motivating Example

We saw in the previous example that multiple states initiated lotteries at different times.

Line plot with several Other Lottery States and Non-Lottery States, as well as three focused states: Ohio, New Mexico, and Maine. Maine is in the middle of the set of lines, while New Mexico is among the highest early on, and Maine is among the highest in May/June.

Plot of percent fully vaccinated rates by U.S. state, Jan.-Sept. 2021

Methods

General: Penalized SCM

The standard SCM had weights \(w_i\) that minimized:

\[ \sum_{k=1}^K v_k \left( X_{1k} - \left(\sum_{i=1}^n X_{0ik} w_i \right) \right)^2, \]

for covariates \(k=1,\ldots,K\).

This can be extended with a penalty term \(\xi\) on a function \(f(w_i)\) to minimize:

\[ \sum_{k=1}^K v_k \left( X_{1k} - \left(\sum_{i=1}^n X_{0ik} w_i \right) \right)^2 + \xi \sum_{i=1}^n f(w_i). \]

The penalty can reduce interpolation (force closer matches to specific units) or reduce discrepancy from an outcome model.

De-Meaned SCM

Title for Doudchenko and Imbens (2016)

Title for Ferman and Pinto (2021)

De-Meaned SCM

Idea

De-mean the pre-treatment data, fit SC to the de-meaned observations, and apply the weights to both the post-treatment time trends and levels.

Incorporates idea of diff-in-diff of focusing on matching trends instead of levels.

Warning

Matching pre-treatment trends may not lead to stable weights going forward.

Different interpretation of weights.

Synthetic DID

Title for Arkhangelsky et al. (2021)

SDID: Procedure

Idea

Incorporate unit weighting of SC with unit fixed effects of DID and time weighting. “Localized” TWFE model.

  1. Compute regularization parameter
  2. Compute regularized, intercept-adjusted/de-meaned SC weights
  3. Compute time weights
  4. Conduct weighted TWFE regression

Augmented Synthetic Control

Title of Ben-Michael et al. (2021)

Idea

De-bias SC estimate using an outcome model for the time series.

Ridge-Adjusted ASCM: Procedure

  1. Compute SC weights
  2. Fit outcome model (i.e., ridge regression) to control data (post-treatment outcomes ~ pre-treatment outcomes)
  3. Get model estimates for all units’ post-treatment outcomes
  4. Find discrepancy between model estimate for treated unit and model estimate for synthetic unit (weighted avg of model estimates for control units)
  5. Add this difference to SC estimator

ASCM: Additional Details

  • If pre-treatment fit is good, discrepancy will be small and adjustment will have little effect

  • Allows for level shift by capturing a consistent discrepancy

  • Can still express as weights of control units, but negative weights now allowed; penalizes discrepancy from SC weights

ASCM: Multiple Periods

Title of Ben-Michael et al., (2022)

Options:

  • Fit separate SCM for each unit

  • Fit SCM on average of treated units

  • Partially pooled SCM: mix of both

  • With intercepts, similar trade-offs to weighted DID approaches

SDID and ASCM: Summary

Title for Krajewski and Hudgens (2024)

Advantages:

  • Allows intercept shift
  • Allow negative weights with some sparsity
  • Improved performance in settings with poor SC fit

SDID and ASCM: Summary

Title for Krajewski and Hudgens (2024)

Disadvantages:

  • Allows extrapolation
  • Loss of interpretation of weights
  • More user degrees of freedom
  • Challenging inference

Generalized Synthetic Control

Title for Xu (2017)

Idea: “Interactive Fixed Effects”

Use control unit data to estimate unit fixed effects and time-varying coefficients. Use these coefficients to estimate treated unit fixed effects and then counterfactuals.

Allows extrapolation and relies on modeling of time series. But can achieve better fit, especially with shift. Allows multiple treated units to be estimated quickly.

Bayesian Structural Time Series Modeling

Title for Brodersen et al. (2015)

Idea

Combine three information sources in state-time model:

  • Bayesian priors on covariate importance
  • Time series modeling of outcome pre-intervention
  • State-space model for treated unit based on controls

BSTS: Summary

  • Allows extrapolation: weights do not sum to 1

  • Incorporates prior information

  • Explicit time series modeling

Title for Bruhn et al. (2017)

Title for Prunas et al. (2021)

Tradeoffs and Interpretations

Validity, Generalizability, and Assumptions

  • Pooling across units enables better fit, but loses unit-specific information

  • Extrapolation allows better pre-treatment fit, but may over-fit or rely on additional assumptions

  • Reducing interpolation is crucial for some settings (non-linear outcomes)

  • Advanced methods allow use of more control series: reduces variance but may introduce bias

Interpretability of Weights

  • A key benefit of SC is its interpretability

  • This is somewhat lost in more advanced approaches

  • The interpretability is tied to justifying the assumptions as well

Questions?